However, independent of the approach used to construct them, the ROM basis functions generally share several qualitative features.
To illustrate this, in Fig. REF  we plot two POD basis functions, \({\varphi }_{1}\)  and \({\varphi }_{10}\) , and two FEM basis functions, \({\phi }_{1}^{h}\)  and \({\phi }_{10}^{h}\) , for a 2D flow past a circular cylinder [1]}.
One can clearly see the significant differences between the POD basis functions (top two plots) and the FEM basis functions (bottom two plots).
Indeed, the POD basis functions have global support (i.e., they can be nonzero over the entire computational domain), whereas the FEM basis functions have local support (i.e., they are one at one mesh point and zero everywhere else).
To further illustrate the different characteristics of the POD basis, in Fig. REF  we plot two POD basis functions, \({\varphi }_{1}\)  and \({\varphi }_{10}\) , for soft tissue modeling [2]}.
Comparing these two POD basis functions with the POD basis functions in the top two plots of Fig. REF , we can clearly see that different physical systems (i.e., the soft tissue in Fig. REF  and the flow in Fig. REF ) yield fundamentally different POD basis functions.
We emphasize that this is in complete contrast with classical numerical methods, such as the FEM.
Indeed, the FEM basis functions are universal basis functions, i.e., they have the same shape (piecewise polynomials and local support) for all the problems.
In contrast, the POD basis functions (and ROM basis functions in general) change their shape when we change the problem.
This can be clearly seen by comparing the top two plots of Fig. REF  with the plots of Fig. REF .
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